Find the mean and variance of the posterior distribution of for the data in question 5

Find the mean and variance of the posterior distribution of θ for the data in question 5 mentioned earlier using the prior you derived in answer to that question by means of the Gibbs sampler ( chained data augmentation).

Question 5

In question 16 in Chapter 2, we supposed that the results of a certain test were known, on the basis of general theory, to be normally distributed about the same mean µ, with the same variance ϕ, neither of which is known. In that question, we went on to suppose that your prior beliefs about (µ,ϕ) could be represented by a normal/chi-squared distribution with 

Find a semi-conjugate prior which has marginal distributions that are close to the marginal distributions of the normal/chi-squared prior but is such that the mean and variance are independent a priori. Now suppose as previously that 100 observations are obtained from the population with mean 89 and sample variance s² = 30. Find the posterior distribution of(µ,∅) Compare the posterior mean obtained by the EM algorithm with that obtained from the fully conjugate prior.

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Vo = 4, So = 350, no 1 1 and 00 = 85.